The study of equations for the recursion coefficients for OPRL or orthogonal polynomials in the unit circle constitutes a subject of current interest. The question of how the form of the weight and its properties, for example to satisfy a Pearson type equation,
translates to the recursion coefficients has been treated in several places, for a review see [1]}. In 1976, Freud [2]} studied weights in \(\mathbb {R}\)  of exponential variation \(w(x)=|x|^\rho \exp (-|x|^m)\) , \(\rho >-1\)  and \(m>0\) . For \(m=2,4,6\)  he constructed relations among them as well as determined its asymptotic behavior. However, Freud did not found the role of the discrete Painlevé I, that was discovered later by Magnus [3]}. For the unit circle and a weight of the form \(w(\theta )=\exp (k\cos \theta )\) , \(k\in \mathbb {R}\) , Periwal and Shevitz [4]}, [5]}, in the context of matrix models, found the discrete Painlevé II equation for the recursion relations of the corresponding orthogonal polynomials. This result was rediscovered latter and connected with the Painlevé III equation [6]}.
In [7]} the discrete Painlevé II was found using the Riemann–Hilbert problem given in [8]}, see also [9]}. For a nice account of the relation of these discrete Painlevé equations and integrable systems see [10]}, and for a survey on the subject of differential and discrete Painlevé equations cf. [11]}. We also mention the recent paper [12]} where
a discussion on the relationship between the recurrence coefficients of orthogonal polynomials with respect to a semiclassical Laguerre weight and classical solutions of the fourth Painlevé equation can be found. Also, in [13]} the solution of the discrete alternate Painlevé equations is presented in terms of the Airy function.
